| L(s) = 1 | + 5·5-s + 21.1·7-s − 67.4·11-s + 2.55·13-s − 126.·17-s + 103.·19-s − 200.·23-s + 25·25-s − 71.4·29-s + 158.·31-s + 105.·35-s + 7.18·37-s + 347.·41-s − 189.·43-s + 585.·47-s + 104.·49-s − 77.2·53-s − 337.·55-s + 200.·59-s + 681.·61-s + 12.7·65-s + 810.·67-s + 515.·71-s + 385.·73-s − 1.42e3·77-s + 209.·79-s + 887.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.14·7-s − 1.84·11-s + 0.0545·13-s − 1.79·17-s + 1.25·19-s − 1.81·23-s + 0.200·25-s − 0.457·29-s + 0.916·31-s + 0.510·35-s + 0.0319·37-s + 1.32·41-s − 0.671·43-s + 1.81·47-s + 0.303·49-s − 0.200·53-s − 0.826·55-s + 0.443·59-s + 1.42·61-s + 0.0243·65-s + 1.47·67-s + 0.861·71-s + 0.618·73-s − 2.11·77-s + 0.298·79-s + 1.17·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.137753925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137753925\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 21.1T + 343T^{2} \) |
| 11 | \( 1 + 67.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.55T + 2.19e3T^{2} \) |
| 17 | \( 1 + 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 71.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 158.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 7.18T + 5.06e4T^{2} \) |
| 41 | \( 1 - 347.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 189.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 585.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 77.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 200.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 681.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 810.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 515.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 385.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 209.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 887.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 548.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.52e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.540540540979471205512359158655, −7.982482534668081806028546961348, −7.36521872314201080867716142096, −6.27567586868156074857304167106, −5.40352117194220232435691405436, −4.87903667465753922498556029228, −3.93744080750870509876976615664, −2.48610120606485606326511153990, −2.07880550143331065301377415272, −0.64153210711821480812153411989,
0.64153210711821480812153411989, 2.07880550143331065301377415272, 2.48610120606485606326511153990, 3.93744080750870509876976615664, 4.87903667465753922498556029228, 5.40352117194220232435691405436, 6.27567586868156074857304167106, 7.36521872314201080867716142096, 7.982482534668081806028546961348, 8.540540540979471205512359158655
